THE ROYAL INSTITUTE OF TECHNOLOGY
MASTER THESIS
The Market Graph
A study of its characteristics, structure & dynamics
Students:
David Jallo
Daniel Budai
December 2010
Supervisors:
Associate Professor Harald Lang
Distinguished Professor Panos M. Pardalos
Abstract
In this thesis we have considered three different market graphs; one solely based on
stock returns, another one based on stock returns with vertices weighted with a liquidity
measure and lastly one based on correlations of volume fluctuations. Research is
conducted on two different markets; the Swedish and the American stock market. We
want to introduce graph theory as a method for representing the stock market in order to
show that one can more fully understand the structural properties and dynamics of the
stock market by studying the market graph. We found many signs of increased
globalization by studying the clustering coefficient and the correlation distribution. The
structure of the market graph is such that it pinpoints specific sectors when the
correlation threshold is increased and different sectors are found in the two different
markets. For low correlation thresholds we found groups of independent stocks that can
be used as diversified portfolios. Furthermore, the dynamics revealed that it is possible
to use the daily absolute change in edge density as an indicator for when the market is
about to make a downturn. This could be an interesting topic for further studies. We had
hoped to get additional results by considering volume correlations, but that did not turn
out to be the case. Regardless of that, we think that it would be interesting to study
volume based market graphs further.
Sammanfattning
I denna uppsats har vi tittat på tre olika marknadsgrafer; en enbart baserad på
avkastning, en baserad på avkastning med likvidviktade noder och slutligen en baserad
på volymkorrelationer. Studien är gjord på två olika marknader; den svenska och den
amerikanska aktiemarknaden. Vi vill introducera grafteori som ett verktyg för att
representera aktiemarknaden och visa att man bättre kan förstå aktiemarknadens
strukturerade egenskaper och dynamik genom att studera marknadsgrafen. Vi fann
många tecken på en ökad globalisering genom att titta på klusterkoefficienten och
korrelationsfördelningen. Marknadsgrafens struktur är så att den lokaliserar specifika
sektorer när korrelationstaket ökas och olika sektorer är funna för de två olika
marknaderna. För låga korrelationstak fann vi grupper av oberoende aktier som kan
användas som diversifierade portföljer. Vidare, avslöjar dynamiken att det är möjligt att
använda daglig absolut förändring i bågdensiteten som en indikator för när marknaden
är på väg att gå ner. Detta kan vara ett intressant ämne för vidare studier. Vi hade
hoppats på att erhålla ytterligare resultat genom att titta på volymkorrelationer men det
visade sig att så inte var fallet. Trots det tycker vi att det skulle vara intressant att
djupare studera volymbaserade marknadsgrafer.
Acknowledgements
We would like to thank Dr. Panos M. Pardalos and Dr. Petraq Papajorgji at the Center
for Applied Optimization, University of Florida for inviting us and giving us the chance
to write our thesis abroad. A special thanks to Dr. Pardalos for his insightful comments,
interesting discussions and professional guidance throughout the process. We would
also like to thank our supervisor Harald Lang at the Institution of Mathematical
Statistics, KTH for making this thesis possible.
Table of Contents
1 Introduction .........................................................................................................................7
2 Theoretical study .................................................................................................................9
2.1 Basic definitions and notations ......................................................................................9
2.2 Weighted graph .............................................................................................................9
2.3 Power-law and scale invariance property ..................................................................... 10
2.4 Clusters, cliques, quasi-cliques and independent sets ................................................... 10
2.5 Liquidity ..................................................................................................................... 12
2.6 Maximum weighted clique- and Maximum weighted independent set problem............ 13
2.7 Constructing the market graph ..................................................................................... 14
2.8 Algorithms .................................................................................................................. 14
2.8.1 NP-hard ................................................................................................................ 14
2.8.2 Heuristic algorithm ............................................................................................... 14
2.8.3 Algorithm for Maximum clique and Maximum weighted clique ........................... 15
2.8.4 Quasi-clique algorithm.......................................................................................... 16
3 Characteristics of the market graph .................................................................................... 18
3.1 Data ............................................................................................................................ 18
3.2 Clustering coefficient .................................................................................................. 18
3.3 Correlation distribution ............................................................................................... 19
3.4 Edge density................................................................................................................ 21
4 Structure of the market graph ............................................................................................. 23
4.1 Maximum clique and Maximum independent set ......................................................... 23
4.2 Maximum weighted clique and Maximum weighted independent set........................... 24
4.3 Maximum quasi-clique and Maximum quasi-independent set ...................................... 25
5 Dynamics of the market graph ........................................................................................... 26
5.1 Data ............................................................................................................................ 26
5.2 Correlation distribution and edge density..................................................................... 27
5.3 Evolution of the market graph over time ...................................................................... 28
6 Conclusion ........................................................................................................................ 34
Bibliography......................................................................................................................... 36
Appendix .............................................................................................................................. 38
I – Cliques and independent sets in the Swedish market .................................................... 38
II – Cliques and independent sets in the American market ................................................. 39
III – Weighted cliques and independent sets in the Swedish market................................... 40
IV – Weighted cliques and independent sets in the American market ................................ 41
V – Quasi-cliques and independent sets in the Swedish market ......................................... 42
VI – Quasi-cliques and independent sets in the American market ...................................... 45
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1 Introduction
In this chapter we introduce the reader to the subject and present the purpose and the
outline of the thesis.
In today‟s world of seemingly endless information, one often faces challenges of
dealing with large sets of data when trying to solve different problems. In many cases,
these massive data sets can be represented by large network structures, or graphs. These
graphs have specific properties and attributes which, if studied properly, provide us with
a lot of information about the applications they portray. Using graphs to represent real
world dynamics is common in many different fields such as military systems and
technology, ecology, telecommunications, medicine and biotechnology, astrophysics,
geographical systems and finance. An example of a data set which can be represented
by a graph is telecommunications traffic data. In that graph, the vertices are telephone
numbers which are connected through edges if a call has been made from one number to
another. Other examples of graph representations are the Internet and the human brain
(1).
In this thesis we will concentrate our efforts on a graph representation of the stock
market, called the market graph. Since the stock market lacks physical connections
between stocks it is not at all obvious how the market can be represented. Nonetheless,
a somewhat intuitive representation of the stock market can be based on the correlations
of stock price movements. Another approach that we will introduce later on in the report
is the correlations of volume fluctuations between stocks. Hence, a market graph can be
constructed by letting each stock be represented by a vertex and let two vertices be
connected by an edge if the correlation coefficient of the stock pair exceeds a pre-
specified threshold. We want to introduce graph theory as a method for representing the
stock market in order to show that one can more fully understand the structural
properties and dynamics of the stock market by studying the market graph.
Being inspired by the article „Statistical analysis of financial networks’ (2), we mainly
want to construct two market graphs; both of which are based on stock returns but with
the difference that one has vertices which are weighted with a liquidity measure. After
having presented the theory that will be used throughout this thesis in chapter 2, we will
analyze the characteristics of both the unweighted and the weighted graph in chapter 3.
In order to get more reliable results we will consider two markets with distinguishable
difference in size, namely the Swedish market and the American market. Chapter 4 is
about the structure of the market graph where we implement means of data mining
using optimization algorithms to find clusters within the graphs. By splitting our data
into different time periods we get more dynamics in our research which is presented in
chapter 5. This way we can extract information about the dynamics of the market graphs
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by studying some of their properties and structures. This chapter will also contain an
analysis of the market graph based on volume correlations since it has to our knowledge
not yet been investigated. We hope that the information obtained from it might help us
understand the market from another point of view. We will conclude the thesis in
chapter 6 where we will summarize our results and have a brief discussion and also
mention how this subject can be studied further.
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2 Theoretical study
Important parts from the field of graph theory as well as the mathematical algorithms
used are presented in this part of the thesis.
2.1 Basic definitions and notations
A graph consists of a nonempty vertex set and an edge set . If is
an edge and are different vertices such that , then and are said
to be adjacent i.e. two vertices are adjacent if they share a common edge. Similarly, two
edges are adjacent if they share a common vertex. The degree of a vertex in a
graph , denoted by , is the number of edges of incident with , each loop counting
as two edges. If is an even number then is said to be an even vertex; if is odd the
vertex is said to be odd, and if then is called an isolated vertex. (3)
For every integer number one can calculate the number of vertices with degree
equal to and then get the probability that a vertex has degree as
, where
is the total number of vertices and the function is referred to as the degree
distribution of the graph (4).
Edge density if defined as the number of edges of a graph divided by the total number
of possible edges in the graph:
, where is the number of vertices of the
graph
2.2 Weighted graph
A weighted graph is a graph in which each vertex is assigned a
nonnegative real number called the weight of . The weight of the graph ,
denoted by , is the sum of the weights of all vertices. Weighted graphs are often
used when practical problems are modeled with means of graph theory. Throughout this
thesis the weights will be represented by the liquidity of the stocks. (3)
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2.3 Power-law and scale invariance property
A quantity obeys a power-law if it is drawn from a probability distribution
or equivalently,
where is a constant parameter of the distribution known as the exponent or scaling
parameter (5).
Power laws have different properties and the main one is perhaps the scale invariance
property
where is a constant. That is, we get a proportional relationship where the original
power-law relation is multiplied by the scaling factor .
In our case, where is the number of nodes with degree , the power-law graph model
is according to (1) defined as,
or equivalently
2.4 Clusters, cliques, quasi-cliques and independent sets
Clusters are groups of data such that objects within a cluster have high similarity in
comparison to one another, but are very dissimilar to objects in other clusters. Since we
in this thesis define vertices as stocks, we consider vertices to be “more similar” the
higher the correlation is between them. Often, one distinguishes to which cluster a
specific vertex belongs by measuring its distance to the rest of the vertices in the cluster;
in this case however, distance can be substituted by correlation. (6)
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A clique is a fully connected graph; i.e. a subset of a graph‟s vertices such that
every two vertices in the subset are connected by an edge (see Figure 2). Considering the
market graph, a clique would characterize a group of highly correlated and interrelated
stocks such as a specific industry. Moreover, a clique is referred to as being maximum if
the graph contains no larger clique and it is called maximal if the clique cannot be
extended to a larger clique. We will in this thesis only focus on maximum cliques (MC).
Another set of interest is the maximum independent set (MIS). It is defined as a set of
vertices no two of which are connected. More formally, it can be depicted as a clique in
the complementary graph . Since this basically is the complete opposite of a clique,
seeing as the vertices are negatively correlated, it is natural to interpret the set as a
possible diversified portfolio. (3)
Figure 1. A graph with 9 vertices.
Quasi-cliques are special kinds of clusters that either have a constraint on minimum
vertex degree or minimum edge density. Hence, a quasi-clique can be defined in two
different ways. With a constraint on minimum vertex degree, a quasi-clique is defined
in the following way: Let be the set of vertices of the subgraph
we wish to find. Then, the set of vertices S is a γ-quasi-clique
i.e. a sub-graph that satisfies the user-specified minimum
vertex degree bound . As a special case, a γ-quasi-clique is a fully
connected graph, or a clique, when γ . (7)
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If on the other hand one would consider a γ-quasi-clique as being a cluster with a
minimum constraint on edge density, the definition would be the same as above but
with the constraints that the graph has to be connected and , i.e. the
number of edges of the graph has to be greater than some number dependent on
and the number of vertices in the graph (8).
Figure 2. A fully connected sub-graph, or clique, within a graph, highlighted in red.
2.5 Liquidity
Most people have an intuitive feeling about what liquidity is but not many can state how
it should be mathematically defined. Linguistically, liquidity can be defined as “the
probability that an asset can be converted into an expected amount of value within an
expected amount of time” (9). In the context of this thesis however, a more suitable
definition of liquidity is “the ability to convert shares into cash (and the converse) at
the lowest transaction costs” (10). There is no consensus in the academic community
exactly how to mathematically quantify the aforementioned definitions, but two
common measures are the bid-ask spread and the turnover rate. The bid-ask spread is
simply the difference between the bid price, the price people are willing to sell a
specific share for at time , and the ask price, the price people are willing to buy a
specific share for at time . The second most common measure, the turnover rate, is
defined as
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The notion is that investors expect a higher rate of return when investing in illiquid
assets since the transaction costs for them are higher than for their more liquid
counterparts. This price premium is evident in markets in the form of the bid-ask spread
where the prices include premiums for immediate buying and immediate selling (11).
Therefore, the bid-ask spread can be regarded as the price one pays for liquidity and
hence, the lower the spread the more liquid the asset is considered to be. The situation is
the converse in the case of the turnover rate, meaning that a higher turnover rate implies
a higher liquidity.
There are many articles that analyze different proxies for liquidity and also the
relationship between liquidity and stock returns, but unfortunately their results are not
conclusive. One of the bigger reasons for that is because the different researchers use
different measures, or proxies, in their attempts to quantify liquidity. (11), (12) and (10)
have all conducted empirical investigations in the matter and found that the bid-ask
spread measure has yielded inconclusive results as a proxy for liquidity while the
turnover rate measure, although not as prevalently used, has led to more stable and
uniform results. This is especially true for quote-driven markets such as the NYSE,
NASDAQ and OMX. In light of that evidence we will in this thesis use the turnover
rate as defined above as our proxy for liquidity.
2.6 Maximum weighted clique- and Maximum weighted independent
set problem
We use the following formulation for the MWC-problem (13):
Max
Subject to:
where:
As a special case, if the graph is unweighted, we set all the weights . The
maximum weighted independent set (MWIS) problem is equivalent to the MWC
problem in the complementary graph and solving it will give the independent set with
largest weight for a given graph.
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2.7 Constructing the market graph
The construction of the market graph is quite simple. We let each vertex represent a
stock and for any pair of vertices and , an edge is connecting them if the
corresponding correlation coefficient , based on the returns of instruments
and , is greater than or equal to a specified threshold . Now, let
denote the price of the instrument on day . Then
defines the logarithm of the return of instrument over the one-day period from
to . The correlation coefficient between instruments and is calculated as
,
where is the average return of stock over days, i.e.
.
2.8 Algorithms
2.8.1 NP-hard
That a problem is NP-hard means, among other things, that it cannot be solved exactly
using in polynomial time. All exact algorithms therefore have exponential runtimes
which makes the solution process that much more difficult (2). However, algorithms for
the maximum clique problem utilize the clique‟s downward closure property, i.e. the
fact that every subset of a clique also is a clique. This piece of information makes it
possible to construct efficient algorithms for the maximum clique problem.
Unfortunately though, the downward closure property does not hold for finding
maximum quasi-cliques (MQC) which means that the algorithms for finding exact
solutions are much less efficient.
2.8.2 Heuristic algorithm
In order to get good starting points for the exact solution method used, we implemented
a fast heuristic method to get approximate solutions to the MC-problem. A heuristic
algorithm will usually not produce the optimal solution. However, a close to optimal
solution is often found within a fraction of the time it takes to run an exact algorithm.
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The heuristic algorithm used in this thesis is called the Vertex Support Algorithm (14).
It is designed to solve the minimum vertex cover problem, which actually is equivalent
to solving the MC- or MIS-problem. A vertex cover is defined as a set of vertices such
that each edge of the graph is incident to at least one vertex in the set. A minimum
vertex cover is a vertex cover of smallest possible size.
The algorithm works in the following way: First we calculate the degree and support of
every vertex in the graph. The support of a vertex is defined as the sum of the degrees of
its neighbors. The vertex with the largest support is then added to the vertex cover and
is subsequently removed from the graph. If two or more vertices have equivalent
maximum support we add the one with the largest degree to the vertex cover. This
continues iteratively and when no edges between the vertices are left we have found our
minimum vertex cover. The MIS or MC, depending on if you look at the graph or its
complement, is then the vertices which are not in the minimum vertex cover.
2.8.3 Algorithm for Maximum clique and Maximum weighted clique
To solve the MC-, MIS-, MWC- and MWIS-problem, Pardalos‟ and Carraghan‟s exact
algorithm was used (15). However, to speed up the algorithm, we implemented a
preprocessing procedure which utilizes the results from the heuristic algorithm. Since
we know that the exact solution for the MC-problem is larger than or equal to the
heuristic result, we can remove all vertices in the graph which has degree smaller than
the heuristic MC size since they obviously cannot be a part of the MC. This will
significantly reduce the problem size and speed up the calculations. The exact algorithm
works in the following way:
We start with one vertex, , and we look for all vertices adjacent to it. When those
nodes are found, we look for all nodes adjacent to the ones we just found that were
adjacent to . This is done iteratively until we find all cliques containing and then we
simply pick out the largest clique containing that vertex and save it as our current best
clique. Next, we remove from the graph and go through the same procedure with the
next node, . Since this algorithm uses brute force to find the MC it would not be
efficient unless some pruning strategies were implemented. The pruning strategies help
to speed up the search in two ways. First of all, every time a new clique is found it is
compared to the current best clique in order to find out if it is larger. If it is, we save the
new clique as the current best clique and discard the old one. Say that we are about to
evaluate a new vertex at some step in the search and our current best clique consists of
ten vertices. If the vertex we are about to evaluate only has 9 neighbors or less, we
know that it cannot be part of a clique larger than our current best one since the largest
clique it can be a part of is of size ten. Therefore we skip that vertex altogether and go
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on to the next. Furthermore, suppose that we have a graph containing 100 vertices, that
we at the moment have searched through 70 of them and that our current best clique is
of size 32. Now we only have 30 vertices left to go through, since we remove every
evaluated vertex from the graph, and hence there is no possibility that we will find a
clique larger than our current best one which is of size 32. Because of that, we will not
look through the last 30 vertices and we have found the MC. The weighted counterpart
of the algorithm works in the same way except that it prunes based on the weight and
not the degree of the remaining vertices. For the interested reader, VB code for the two
algorithms can be found on the internet (16).
2.8.4 Quasi-clique algorithm
The exact optimization problem for finding MQC is very difficult, as well as
computationally challenging, to solve. The main problem is one concerning memory
and computational time. To be able to run the optimization on a single desktop
computer one would have to write a very efficient program, where as little memory as
possible is needed in every step of the calculation. In addition, the time it would take to
find a globally optimal solution is too long because of two reasons; first of all since the
problem is more complex than the MC-problem, which is NP-hard, and secondly
because it doesn‟t satisfy the same closure property as the MC-problem does. Therefore,
a greedy randomized adaptive search procedure (GRASP) with a local search algorithm
will be used instead of an exact algorithm. GRASP is an iterative method that constructs
a random solution, i.e. a clique, at each iteration, and then searches for a locally optimal
solution in the neighborhood of the created clique (17). This way one cannot know how
good QC one obtains. However, this is not a problem since the goal is not to find exact
solutions but rather to identify different sectors and find larger independent sets.
In the beginning of the algorithm, a vertex is randomly chosen from a list of vertices
that all have degrees greater than some threshold. This vertex will serve as the start of
the clique. The next vertex to be added to the clique is chosen based on a similar list
wherein all vertices are adjacent to the first chosen vertex whilst their degrees are
greater than some new threshold. This procedure is repeated until no more candidates
can be found and then we have a found a solution. Now we implement a local search
procedure in order to improve the solution. The local search creates a better solution by
randomly choosing a vertex from the previously obtained solution, removing it from the
clique, and then adding two or more new vertices that are connected to all vertices in the
remaining clique. This continues as long as it is possible to find such vertices that, if
they are removed, can be replaced by two or more other vertices to improve the
solution.
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The only difference between our algorithm for MQC and the one explained above for
cliques is that the constraint on the vertices one adds is relaxed. Instead of demanding
that they are connected to all the other vertices it is sufficient that they are connected to
at least of the vertices in the clique. This will furthermore guarantee that
the new solution‟s edge density is at least
.
Since there is a constraint on the degree of each vertex, instead of on the edge density,
no undesirable QC with high edge density but including vertices with only one
connection to the rest of the QC will be found. Therefore, we are ensured to only find
MQC of good quality.
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3 Characteristics of the market graph
In this part of the thesis we explore and analyze some traits of the market graph.
3.1 Data
In order to present some characteristics of the market graph and study its dynamics we
have used stock returns from October 20, 2008 to October 15 2010. Research will be
conducted on two different markets; the Swedish stock market OMX and the American
stock market consisting of NASDAQ, AMEX and NYSE. This will give us the
possibility to compare the different markets as well as get better and more reliable
results. 266 stocks have been collected for the Swedish market for 500 consecutive
trading days and 5700 stocks for the American market for 502 consecutive trading days,
the two additional days being due to differences in holidays. Although there is a loss in
the amount of American securities, since data was not available for some of them, we
believe that we have enough data to get reliable and consistent results.
3.2 Clustering coefficient
The clustering coefficient is a probability measure that quantifies the probability that the
nodes adjacent to a single node are connected. In other words, it gives us a measure of
how well nodes in a graph tend to cluster together and thus, how well connected the
neighborhood of a node is. Let us look at node which is of degree . Then we get the
clustering coefficient for that node by taking the ratio of the number of edges that
actually exist between its neighbors and the total number of edges that
could exist in the neighborhood of , i.e.
The entire graph‟s clustering coefficient is simply the mean of the individual clustering
coefficients of the nodes that have degree greater than two.
When calculating the clustering coefficient for different values of the correlation , we
found that the clustering coefficient is higher for large and positive in comparison to
small and negative in the complementary graph, where the clustering coefficient
turned out to be very close to 0. We suggest that this is a sign of globalization, meaning
that more and more stocks are dependent on each other and that the market movements
are less random. For instance, with , the Swedish market has almost the same
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edge density in the original graph as in the complementary graph with .
However, the corresponding values of the clustering coefficients are and
. This result is analogous in the American market graph where for
and for . Consequentially, we expect to find significantly
larger MC than MIS in both market graphs. This result corresponds to the findings in
(2). The fact that the clustering coefficient is much higher than the edge density is a
typical characteristic for power-law graphs.
3.3 Correlation distribution
As one of the characteristics of the market graph, the correlation distribution provides
information about how the stocks are correlated to one another, thus telling us what type
of market structure we are dealing with. Figure 3 shows the correlation distribution of
the Swedish stock market where the red curve is a normal distribution fitted to the data.
Obviously, the correlations between stocks at OMXS are not normally distributed. The
data lacks symmetry and the heavy tail on the right will not be encompassed by a
normal curve. Moreover, the mean value is and the standard deviation
is . With that in mind and the fact that the correlation of most stocks are greater
than zero, i.e. the stock prices tend to move in the same direction, we get yet another
indication that the modern stock market is affected by globalization.
Figure 4 is the corresponding plot for the American stock market with and
In contrast to OMXS; both tails of the correlation distribution of the
American stocks are almost entirely covered by the fitted normal distribution. However,
the shape of the correlation distribution and the shape of the normal distribution do not
match. Thus, a normal fit is not appropriate. Similar to the Swedish market, the
American stocks also mainly exhibit positive correlations which further corroborate the
theory of increased globalization.
By studying the evolution of the correlation distribution of the American market graph
over time, one can show that it remains stable. Consequently, the degree distribution
will remain stable over time and a plot can be approximated by a straight line (in a
logarithmic scale), which means that it can represent the power-law distribution. For a
more stringent analysis see (4). We will later on in the report study the evolution of the
Swedish market graph over different time periods.
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Figure 3. Distribution of correlation coefficients in the Swedish stock market with a fitted
normal distribution.
Figure 4. Distribution of correlation coefficients in the American stock market with a fitted
normal distribution.
21
3.4 Edge density
Edge density is a ratio obtained by dividing the numbers of edges in a graph with the
maximum possible number of edges, where is the number of vertices in
the graph. Changing the value of the correlation threshold will affect the edge density,
and by doing so one can construct market graphs with different degrees of correlation
between stocks. This can be used to alter sizes of cliques and independent sets in a
graph. Figure 5 shows the edge density for the Swedish stock market for different
values of the correlation threshold. It is clear that the edge density decreases with
increasing threshold values. This result is not surprising since we expect to find fewer
stocks that behave similarly as we increase the correlation threshold. Also, higher edge
density is linked to lower correlation between stocks, which is in line with the notion
that a portfolio with a larger amount of stocks is better diversified. Figure 6 shows the
edge density of the American stock market and it is easy to see that it almost has the
exact same shape as the Swedish. One can expect similar shapes for any stock market in
the world.
In (4) the authors studied the edge density and its change during different consecutive
time periods. By setting the value of the correlation threshold to , they made sure
that they got edges that corresponded to those stocks which were significantly
correlated with each other. It turned out that the edge density was approximately 8.5
times higher in the last period than the first which, according to the authors, was an
indication of the increasing globalization of the modern stock market.
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Figure 5. Edge density of the Swedish market graph for different values of the correlation
threshold.
Figure 6. Edge density of the American market graph for different values of the correlation
threshold.
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4 Structure of the market graph
In this chapter we utilize the earlier described algorithms in order to find differences
and similarities between stocks in the graphs.
4.1 Maximum clique and Maximum independent set
As we already have mentioned, the MC is the largest cluster in which all nodes are
connected to every other node, thus making it a complete graph. The MIS is the
corresponding complete graph in the complementary market graph. Keeping in mind
that we are looking at return correlations between stocks, the MC will represent the
maximum number of stocks whose price fluctuations exhibit similar behavior.
Correspondingly, the MIS represents the maximum set of stocks whose price returns are
the most uncorrelated, and thus constitutes the largest diversified portfolio.
In the previous chapter we mentioned that it is easier to find a MC in the original graph
than a MIS in the complementary graph. By looking in Appendix I and Appendix II,
one quickly realizes that this is also the case. The MIS for are
smaller than the MC for . Moreover, the MIS size becomes even smaller
for which is consistent with the results we got from Figure 3 and Figure 4,
namely that globalization has a strong affect on the market. The fact that the MIS are
small, and hence contain too few stocks to choose from when considering building a
portfolio, we are led to search for alternate methods which can assist us in finding good,
diversified portfolios. This method will be explored in chapter 4.3.
Comparing Appendix I to Appendix II, we clearly see that the Swedish market yields
smaller MIS than the American market. One way to decrease unsystematic risk is to
hold a portfolio consisting of many uncorrelated stocks. Therefore, it is favorable to
invest in the MIS in markets of larger size.
At for the Swedish market graph, the MC includes stocks from the industrial
and the material sectors. With higher values of , one can expect to get a clique
consisting of stocks from only one sector. However, the result is still satisfying since the
industrial and the material sector are highly correlated. This is quite obvious since the
manufacturing industry depends on the companies supplying their materials in order to
function and conduct business properly. Another interesting observation is that three
financial companies appear in the same MC. This is due to the large positions that the
financial companies have in the other companies from the same clique. Consequently at
the MC will neither include the financial company INDU nor will it include
the industrial companies SAND and VOLV, the two of which INDU has large positions
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in. The same behavior is found in the American market. According to Appendix B, the
MC at only consists of companies from the basic materials industry, more
specifically silver and gold companies. Unlike the Swedish market, these companies do
not emerge in other cliques and must therefore be very strongly correlated which is
something one can anticipate from both silver and gold securities.
An interesting observation is that as decreases, the algorithm either adds stocks from
sectors already existing in the MC or stocks belonging to companies that in some other
way are highly dependent on the ones already in the MC. An interesting difference
between the two markets is that the cliques in the Swedish market are based on some of
the biggest companies while the cliques in the American market are built strictly around
specific sectors.
4.2 Maximum weighted clique and Maximum weighted independent
set
By adding weights to the stocks we get solutions to the MWC- and MWIS-problem
which not only considers the price fluctuations between the stocks, but also the
liquidity. This will provide us with information about the stock market from another
perspective that we can compare to the unweighted case.
The cliques in the weighted case behave very similarly to the unweighted ones except
for a few notable differences. Instead of pinpointing gold and silver companies, the
algorithm for the MWC-problem generates cliques consisting of market indices for
in the American market. This means that not only are the indices
highly correlated with each other, but they are also highly liquid. The Swedish market
on the other hand is unaltered since we only included stocks from OMX. All stocks
included in the Swedish MC and MWC are, unsurprisingly, from OMX Large Cap.
Thus, they are the most correlated and most liquid stocks at the same time. We
furthermore found that the industrial and the material sectors appear in the weighted as
well as in the unweighted case and we therefore draw the conclusion that these stocks
are the most correlated as well as the most liquid. Moreover, it is preferable to choose
diversified portfolios from the weighted graphs since their liquidity risk is lower, even
though their sizes are a bit smaller than the diversified portfolios in the unweighted case
(see Appendix III and Appendix IV).
25
4.3 Maximum quasi-clique and Maximum quasi-independent set
Since the MIS for are small in both markets, we look for MQIS in order to find
larger independent sets. By calculating MQIS, we reduce the requirements in the sense
that we no longer demand complete graphs. Stocks can be a part of MQIS even though
they are not connected to all other stocks in the same MQIS. Thus, we can expect larger
MQIS for the price of less diversification. For instance, in the American graph at
and , a MQIS consisting of 21 stocks, i.e. about 60 % larger than the
corresponding MIS, is found (see Appendix V). Also, a MQIS in the Swedish market
graph at and will generate a quasi clique consisting of 33 stocks,
which is a significantly larger diversified portfolio than in the earlier cases (see
Appendix VI). However, each stock within the MQIS only needs degree
and not 32 in order to be accepted as a part of the QIS. Investing in such
a portfolio would be riskier since the information about how the stocks are correlated
i.e. exactly how diversified the portfolio really is, is somewhat incomplete. The MQC
does however provide us with useful information which, if utilized appropriately,
increases our chances of building a large, well diversified portfolio.
26
5 Dynamics of the market graph
Here we study how the interaction between stocks in the market change as time goes by.
5.1 Data
Earlier we studied and analyzed static market graphs but will now shift our focus to how
some of the previously studied features change over time. The hope is to learn more
about how those features evolve and what, if anything, that says about the market as
time goes by. The data used for this part of the study is the same as the Swedish data
used earlier but with a longer time span, namely between April 20, 2008 and October
15, 2010. We then split our data into four equally large periods which resulted in four
data sets, each consisting of 155 observations of daily returns. Out of those sets of data
we constructed four market graphs and computed their correlation distribution and edge
density. This can be seen in Figure 7 and in Figure 8.
Figure 7. Price correlation density for the Swedish stock market for different time periods.
27
Figure 8. Edge density for the Swedish stock market for different time periods.
5.2 Correlation distribution and edge density
From Figure 7 and Figure 8 we instantly see that the four periods are quite different.
Even though periods 1 and 3 appear to be fairly similar it is obvious that they
significantly differ from the other two periods. Most importantly, the right tails in
periods 1 and 4 (see Figure 7) are greater than for the other two periods and that has an
impact on the edge density as well (see Figure 8). This indicates that one would find
larger cliques and smaller independent sets if one was looking in period 1 instead of in
any other period. Furthermore, we learn that a single market graph constructed using
combined data from all the four periods, as was done earlier, is considerably different
from the four market graphs constructed with the periodically divided data. Since the
market and its structure constantly changes it is important to visualize and keep in mind
what impact that can have on the final results. Lastly, this provides more evidence
strengthening the hypothesis that negative returns tend to correlate more than positive
returns. This is apparent since the first period in our data consists of the last six months
of the downturn in the recent financial crisis and that period is also by far the most
positively correlated and has the highest edge density of all four measured periods.
28
5.3 Evolution of the market graph over time
Now that we have seen how much the market graph can vary depending on the choice
of time period, we think it would be interesting to make a more in-depth study of how
its characteristics evolve during our two and a half years of data. In order to do so we
create market graphs for all 100-day and 20-day periods in our data, i.e. one graph for
day 1-100, one for 2-101, one for 3-102 etc. and we analyze how their properties change
over time.
We begin by considering the evolution of the market graph with a correlation threshold
of 0.5 and 100-day intervals. We calculate the mean correlation coefficient, edge
density, clique number and clustering coefficient for each period and compared them to
the OMXSPI, which is representative of our data. This is done in order to find out if we
can acquire new knowledge about the market.
Figure 9. Mean correlation in the Swedish market graph plotted vs. the OMXSPI for continuous
100-day periods.
29
Figure 10. Edge density of the Swedish market graph plotted vs. the OMXSPI for continuous
100-day periods.
Figure 11. Clique number generated from the Swedish market graph plotted vs. OMXSPI for
continuous 100-day periods.
30
The first observation we make is that the green curves in Figure 9, Figure 10 and Figure
11 look quite similar whilst being negatively correlated with the OMXSPI. We find that
this is true since the edge density and the mean correlation have a correlation of about
0.96 with each other and –0.5 with the index. It is also interesting to see that the clique
number follows the pattern of the mean correlation and the edge density. This is because
the clustering coefficient never drops below 0.65 for the entire period, and as explained
earlier, a high clustering coefficient leads to graphs with denser clusters since new
edges tend to be added to already dense areas of the graph. However, even though we
find that the edge density and mean correlation is strongly negatively correlated with the
market, we cannot really use the information from Figure 9, Figure 10 and Figure 11 for
anything useful since the 100-day period is far too long in order for us to be able to
detect swift changes in market movements. We will therefore divert our attention to the
shorter time period we have intended to study, which is the 20-day period with
correlation threshold 0.2. The reason we chose to lower the correlation threshold in the
20-day period case is because we wanted a higher edge density in order to get more
observations and thus better results. Still, 20-day correlations are not entirely reliable
because of the small number of observations but if one wants to be able to catch quick
market movements using correlation we believe that the only way is to shorten the time
period.
Figure 12. Edge density of the Swedish market graph plotted vs. the OMXSPI for continuous
20-day periods.
31
Figure 13. Change in edge density larger than 0.05 of the Swedish market graph plotted vs. the
OMXSPI for continuous 20-day periods.
Comparing the 20- day edge density (see Figure 12) with the analogous one for the 100-
day period (see Figure 10) we find that the latter is much more volatile. It is however
difficult to draw any more conclusions by only studying Figure 10 which is why we
calculated the days on which the edge density increased by more than 5 percentage
points (see Figure 13). Interestingly, with only two exceptions, it seems as if the edge
density only increases by 5 percentage points or more when the market is about to make
a sharp downturn. This implies that there is a possibility to use the daily absolute
change in edge density as an indicator for when the market is about to go down.
When significant upward or downward jumps occur in the market it is natural to expect
that, just as in the case with return correlations, the correlation between different assets‟
trading volume increase at the same time. To test this hypothesis we construct a market
graph using volume correlations instead of price correlations to see if we get similar
results. In contrast to the results for the return based market graph, Figure 14 and Figure
15 clearly indicate that the volume correlation and the edge density for the different
periods are very similar to one another. Moreover, we can see in Figure 16 that the
peaks of the absolute change in edge density do not pinpoint any distinct downturns in
the market index in the same way they do for the price based market graphs.
32
Figure 14. Volume correlation density in the Swedish market graph for different time periods.
Figure 15. Edge density for the Swedish market graph based on volume correlations for
different time periods.
33
Figure 16. Change in edge density larger than 0.05 of the Swedish market graph based on
volume correlations plotted vs. the OMXSPI.
34
6 Conclusion
The results are summarized and topics for future research are proposed.
In this thesis we have considered three different market graphs; one solely based on
stock returns, another one based on stock returns with vertices weighted with a liquidity
measure and lastly one based on correlations of volume fluctuations. Research was
conducted on two different markets; the Swedish stock market OMX and the American
stock market consisting of NASDAQ, AMEX and NYSE.
We found that the clustering coefficient, in both market graphs, was higher for large
positive correlations in comparison to small and negative correlations in the
complementary graphs. This implies that the MC we found were larger than the MIS
which is an effect of globalization. Further, the correlation distributions turned out to
lack symmetry and have heavy tails to the right, i.e. the correlations of most stocks are
greater than zero. This is yet another sign of the increased globalization making it
harder to find diversified portfolios with time.
Solving the MC-problem, we managed to pinpoint specific sectors for higher values of
the correlations threshold. For the Swedish market we ended up with the industrial and
the material sector, two industries that are highly dependent on each other. The basic
material industry, more specifically silver and gold companies, was pinpointed for the
American market graph. When we decreased the correlation threshold we found that the
algorithm mainly added stocks from the same sector. One of the differences between the
two markets was that the cliques in the Swedish market were based on some of the
biggest companies while the cliques in the American market were built strictly around
specific sectors. Also, in both market graphs, the MIS we found were significantly
smaller than the MC.
The cliques for the weighted case behaved very similarly to the unweighted except for a
few notable differences. Instead of pinpointing gold and silver companies, the algorithm
for the MWC-problem generated cliques consisting of market indices in the American
market, telling us that the indices are highly correlated at the same time as they are very
liquid. The Swedish market on the other hand turned out to be unaltered since we only
included stocks and no indices from OMX. If one ought to invest in a diversified
portfolio, it is clearly preferable to do so from the weighted graphs due to their lower
liquidity risk. In order to increase the size of the diversified portfolio we also calculated
MQIS which gave us independent sets consisting of a larger number of stocks. The
price we had to pay was that such a portfolio would be riskier since the information
about how the stocks are correlated, i.e. how well diversified the portfolio really is, is
somewhat incomplete.
35
When we split the data into four equally long time periods we found that the price
correlations and edge density were quite different. We discovered that one would find
larger cliques and smaller independent sets by looking in period 1 instead of in any
other period. Furthermore, we learnt that a single market graph constructed using
combined data from all four periods is considerably different from the four market
graphs constructed with the periodically divided data. This provided more evidence
strengthening the hypothesis that negative returns tend to correlate more than positive
returns.
The 20-day edge density presented a quite interesting behavior. It seems as if it
increases by 5 percentage points or more every time the market is about to make a sharp
downturn. This implies that there is a possibility to use the daily absolute change in
edge density as an indicator for when the market is about to go down. Obviously this
has to be studied further but it is nonetheless an interesting result which maybe even can
become a tool in predicting market declines.
Using volume correlations instead of price correlations did not add any new results.
Unlike return correlations, the volume correlation distribution and its edge density are
very similar to each other for the different time periods. Moreover, we found that the
peaks of the change in edge density do not pinpoint any distinct declines in the market
index. We had hoped to get additional results by considering volume correlations, but
that did not turn out to be the case. Regardless of that, we think that it would be
interesting to study volume based market graphs further and perhaps try to find
interesting properties which do not exist in the price based graph.
36
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38
Appendix
I – Cliques and independent sets in the Swedish market
Correlation
threshold
Number of
stocks
Stocks
– 0.05
3 ACOM ICTA-B SAS
0
5 ACAN-B DGC NOVE RROS SAS
0.05
14 ARTI-B BALD-B CEVI FEEL GVKO-B KARO
MSON-B NOTE NSP-B ORTI-B PSI-SEK RROS
SAEK WAFV-B
0.2
76
AAK ABB ALFA ALIV-SDB AOIL-SDB ASSA-B
ATCO-A ATCO-B AZA BBTO-B BEF-SDB BEGR
BINV BOL CAST ECEX ELUX-B ERIC-A ERIC-B
FABG GETI-B HEXA-B HOGA-B HOLM-B HUSQ-A HUSQ-B IJ INDU-A INDU-C INVE-A INVE-B
JM KINV-B KLED KLOV KNOW LIAB LUMI-
SDB LUND-B LUPE MEDA-A MIC-SDB MTG-B
NCC-A NCC-B NDA-SEK NISC-B NOBI ORES
ORI-SDB PEAB-B RATO-B SAAB-B SAND SCA-A
SCA-B SCV-A SCV-B SEB-A SEB-C SECU-B SHB-
A SHB-B SKA-B SKF-A SKF-B SSAB-A SSAB-B
STE-R SWED-A TEL2-B TLSN WIHL VNIL-SDB
VOLV-A VOLV-B
0.3
54
ABB ALFA ALIV-SDB AOIL-SDB ASSA-B ATCO-
A ATCO-B BEGR BOL CAST ECEX ELUX-B
FABG GETI-B HEXA-B HOLM-B HUSQ-A HUSQ-
B IJ INDU-A INDU-C INVE-A INVE-B JM KINV-B
KLED LIAB LUND-B LUPE MTG-B NCC-B NDA-
SEK ORI-SDB PEAB-B RATO-B SAND SCA-B
SCV-A SCV-B SHB-A SHB-B SKA-B SKF-A SKF-
B SSAB-A SSAB-B STE-R SWED-A TEL2-B TLSN
WIHL VNIL-SDB VOLV-A VOLV-B
0.4
38
ABB ALFA AOIL-SDB ASSA-B ATCO-A ATCO-B
BEGR BOL ECEX ELUX-B HEXA-B INDU-A
INDU-C INVE-A INVE-B JM KINV-B LUPE MTG-
B NCC-B NDA-SEK PEAB-B RATO-B SAND SCA-
B SCV-B SHB-A SHB-B SKA-B SKF-B SSAB-A
SSAB-B SWED-A TEL2-B TLSN VNIL-SDB VOLV-A VOLV-B
0.5
25
ABB ALFA ASSA-B ATCO-A ATCO-B BOL
ELUX-B INDU-A INDU-C INVE-A INVE-B JM
KINV-B LUPE MTG-B NCC-B NDA-SEK SAND
SCV-B SKA-B SKF-B SSAB-A SSAB-B TEL2-B
VOLV-B
0.6
15
ABB ALFA ATCO-A ATCO-B BOL INDU-C
INVE.-A INVE-B KINV-B SAND SKF-B SSAB-A
SSAB-B VOLV-A VOLV-B
0.7
8
ALFA ATCO-A ATCO-B INVE-A INVE-B SKF-B
SSAB-A SSAB-B
39
II – Cliques and independent sets in the American market
Correlation
threshold Number of stocks
Stocks
– 0.05
6
BNC NEFOI HMNA MEDQ SNFCA
VSCP
0
12
ALLB AMTC ARCW CO DD-PA GJJ
IMS QADI RGCO SSE UNAM WBNK
0.05
35
AERL ANX BDCO BDL CALL CFBK
CO EDCI EDS FFDF GAI GJK GJL
GLOI GSLA INV JCDA KGJI LSBI
NBXH NFEC NFSB NPBCO OGXI
PDEX PSBH RDIB ROIAK RPTP
SKH SPRO UBOH ULCM WWIN
ZANE
0.65
57
ACC AIV AKR AMB ARE AVB BFS
BRE BXP CLI CPT DCT DEI DLR
EGP ELS EPR EQR ESS EXR FRT
FSP HCN HCP HIW HME HR HST
IRC JLL KIM KRC LRY MAA NHP
NNN O OFC OHI PCH PCL PKY PPS
PRFZ PSA REG RYN SKT SNH SPG
SSS TCO UDR VNO WRE WRI VTR
0.7
41
ACC AMB ARE AVB BFS BRE BXP
CLI CPT DCT DEI DLR ELS EPR
EQR FRT HCN HCP HIW HME HR
KIM KRC LRY MAA NNN O OFC
OHI PCH PSA REG SNH SPG SSS
TCO UDR VNO WRE WRI VTR
0.75
31
AVB BRE BXP CLI CPT DCT ELS
EQR FRT HCN HCP HIW HME HR
KIM LRY MAA NHP NNN O OHI
PCH PSA REG RYN SPG TCO UDR
VNO WRE WRI
0.8
16
BRE BXP CLI CPT ELS EQR FRT
HCP HIW LRY NNN O PSA REG
SPG VNO
0.85
5 ABX AEM AUY GG KGC
40
III – Weighted cliques and independent sets in the Swedish market
Correlation
threshold Number of stocks
Stocks
– 0.05
2 ENRO ORTI-A
0
2 HEBA-B LUMI-SDB
0.05
12 ARTI-B BALD-B DORO ENRO HQ LUXO-
SDB MSC MULQ ORTI-A RROS RTIM SAS
0.2
76
AAK ABB ALFA ALIV-SDB AOIL-SDB
ASSA-B ATCO-A ATCO-B AZA BBTO-B
BEF-SDB BEGR BOL CAST ECEX ELUX-B
ERIC-A ERIC-B FABG GETI-B HEXA-B
HOGA-B HOLM-B HUSQ-A HUSQ-B IJ
INDU-A INDU-C INVE-A INVE-B JM
KINV-B KLED KLOV KNOW LIAB LUMI-
SDB LUND-B LUPE MEDA-A MIC-SDB MTG-B NCC-A NCC-B NDA-SEK NISC-B
NOBI ORES ORI-SDB PEAB-B RATO-B
SAAB-B SAND SCA-A SCA-B SCV-A SCV-
B SEB-A SEB-C SECU-B SHB-A SHB-B
SKA-B SKF-A SKF-B SSAB-A SSAB-B
STE-R SWED-A TEL2-B TLSN WIHL
VNIL-SDB VOLV-A VOLV-B
0.3
52
ABB ALFA ALIV-SDB AOIL-SDB ASSA-B
ATCO-A ATCO-B BEGR BOL ECEX
ELUX-B FABG GETI-B HEXA-B HOLM-B
HUSQ-A HUSQ-B IJ INDU-A INDU-C
INVE-A INVE-B JM KINV-B KLED LIAB
LUMI-SDB LUPE MTG-B NCC-B NDA-
SEK ORI-SDB PEAB-B RATO-B SAND
SCA-B SCV-A SCV-B SECU-B SHB-A SHB-
B SKA-B SKF-A SKF-B SSAB-A SSAB-B
SWED-A TEL2-B TLSN VNIL-SDB VOLV-
A VOLV-B
0.4
34
ABB ALFA AOIL-SDB ASSA-B ATCO-A
ATCO-B BEGR BOL ECEX ELUX-B
HEXA-B INDU-A INDU-C INVE-A INVE-B
JM KINV-B LUMI-SDB LUPE MTG-B NCC-
B NDA-SEK PEAB-B RATO-B SAND SCA-
B SKA-B SKF-B SSAB-A SSAB-B SWED-A TEL2-B VNIL-SDB VOLV-A VOLV-B
0.5
25
ABB ALFA ASSA-B ATCO-A ATCO-B BOL
ELUX-B INDU-A INDU-C INVE-A INVE-B
JM KINV-B LUPE MTG-B NCC-B NDA-
SEK SAND SCV-B SKA-B SKF-B SSAB-A
SSAB-B TEL2-B VOLV-B
0.6
15
ABB ALFA ATCO-A ATCO-B BOL INDU-C
INVE.-A INVE-B KINV-B SAND SKF-B
SSAB-A SSAB-B VOLV-A VOLV-B
0.7
8
ALFA ATCO-A ATCO-B INVE-B SAND
SKF-B SSAB-A SSAB-B
41
IV – Weighted cliques and independent sets in the American market
Correlation
threshold
Number of stocks
Stocks
– 0.05
4 CLRO REE SCKT TORM
0
9
ALRN CBIN FCAP MTSL OPTC
PKT RITT SCKT TORM
0.05
25
AMIE BTC BWOW CLSN CNYD
COBK CZFC DJSP EONC GJI ISRL
KENT KRY KSW LEO LONG LSBI
NMRX RITT SAVB TORM TRNS TZF USATP ZAGG
0.65
43
AA ACI ACWX ADRE AKS APA
ATW BTU BUCY CAM CNQ CNX
COP DRQ ECA FCX HAL JOYG
MEE MRO MUR NBL NBR NE
NOV OII OIS OXY PBR PDE PRFZ
PTEN QQQQ RDC SCCO SLB SU TLM UNT VALE WFT WLT
0.7
17
ACI AKS ATI BTU BUCY CLF
CNX FCX JOYG MEE NUE QQQQ
SCCO STLD VALE WLT X
0.75
6
ADRE ONEQ PRFZ QQEW QQQQ
QTEC
0.80
4 ONEQ QQEW QQQQ QTEC
0.85
2 QQQQ QTEC
42
V – Quasi-cliques and independent sets in the Swedish market
Correlation
threshold
Degree
threshold
Number of
stocks Stocks
0.05
0.5 59
ACAN-B AERO-B ARTI-B
ATEL AZN BALD-B BTS-B
CATE CEVI DAG DGC
DIOS DORO DUNI DV
ELEC ELGR-B ELUX-A
FEEL GVKO-B HAV-B HQ
ICTA-B ITAB-B KABE-B
KARO LJGR-B LUXO-SDB
MOBY MSC-B MSON-A
MSON-B MTG-A MTRO-
SDB-A MTRO-SDB-B MULQ NAXS NCAS NOTE
NOVE OEM-B ORTI-A
ORTI-B PHON PREC PROB
PSI-SEK RROS RTIM-B
SAEK SAGA-PREF SAS
SOBI TILG TRAC-B WAFV-
B VITR VRG-B XANO-B
0.05
0.6 46
AERO-B ARTI-B AZN
BALD-B BTS-B CATE CEVI
DAG DGC DORO ELEC
ELGR-B FEEL GVKO-B
HAV-B HEBA-B HMS HQ
ICTA-B LAMM-B LUXO-
SDB MOBY MODL MSC-B
MSON-A MSON-B MTG-A
MTRO-SDB-A MULQ NAXS
NCAS-B NSP-B ORTI-A
ORTI-B PHON PROB PSI-SEK RROS RTIM-B SAEK
SAGA-PREF SAS SOBI
TILG WAFV-B VITR
0.05
0.7 33
ARTI-B BALD-B CATE
CEVI DORO ELGR-B FEEL
GVKO-B HMS HQ ICTA-B KARO MSC-B MSON-B
MTG-A MTRO-SDB-A
MULQ NOTE NSP-B ORTI-
A ORTI-B PHON PREC
PROB PSI-SEK RROS RTIM-
B SAEK SAS SOBI TILG
TRAC-B WAFV-B
0.05
0.8 26
ARTI-B BALD-B CEVI
DORO FEEL GVKO-B HQ
ICTA-B KARO LUXO-SDB
MSC-B MSON-B MULQ
NOTE NSP-B ORTI-A ORTI-
B PHON PSI-SEK RROS
RTIM-B SAEK SAS SOBI
TILG WAFV-B
0.05
0.9 18
ARTI-B BALD-B DORO
ENRO ICTA-B KARO
43
LUXO-SDB MSC-B MULQ NOTE ORTI-A ORTI-B
PHON PREC RROS RTIM-B
SAEK SAS
0
0.5 14
DGC HEBA-B HOLM-A
ICTA-B MOBY MSC-B
MSON-B NSP-B ORTI-A
ORTI-B PSI-SEK RROS SAS TRAC-B
0
0.6 8
ARTI-B CEVI HAV-B
MSON-B ORTI-A SAS SOBI
WAFV-B
0
0.7 8
ACAN-B DGC MSON-B
ORTI-A ORTI-B PSI-SEK
RROS SAS
44
Correlation
threshold
Degree
threshold
Number of
stocks Stocks
0.6
0.5 23
ABB ALFA ASSA-B
ATCO-A ATCO-B BOL
ELUX-B INDU-A INDU-C
INVE-A INVE-B KINV-B
NCC-B RATO-B SAND SCV-B SHB-A SKA-B
SKF-B SSAB-A SSAB-B
VOLV-A VOLV-B
0.6
0.6 19
ABB ALFA ATCO-A
ATCO-B BOL INDU-A
INDU-C INVE-A INVE-B KINV-B NDA-SEK SAND
SHB-A SKA-B SKF-B
SSAB-A SSAB-B VOLV-A
VOLV-B
0.6
0.7 19
ABB ALFA ATCO-A
ATCO-B BOL INDU-A INVE-A INVE-B KINV-B
NCC-B RATO-B SAND
SCV-B SKA-B SKF-B
SSAB-A SSAB-B VOLV-A
VOLV-B
0.7
0.5 11
ALFA ATCO-B INDU-C INVE-A INVE-B KINV-B
SAND SKA-B SKF-B
SSAB-A SSAB-B
45
VI – Quasi-cliques and independent sets in the American market
Correlation
threshold
Degree
threshold
Number of
stocks Stocks
0
0.6 21
BDL BTI CART CIZN CYCCP EEI EMCF FFDF
HAVNP IVA KENT KGJI
MYF PBHC PCBS PFIN
RDIB RIVR SGRP TORM
TRCI
0
0.7 14
ADTN BDL CLRO CWBC EOSPN GIA IVA KGJI
LPTH NRB SGRP TRCI
VMEDW WWIN
0.8
0.5 27
BRE BXP CLI CPT DCT
ELS EQR FRT HCN HCP
HIW HME HR LRY MAA NNN O PCH PSA REG
SNH SPG TCO UDR VNO
WRE WRI
0.8
0.6 24
BRE BXP CLI CPT ELS
EQR FRT HCN HCP HIW
HME HR LRY MAA NNN O PCH PSA REG SNH SPG
TCO UDR VNO
0.8
0.7 21
BRE BXP CLI CPT EQR
FRT HCP HIW HME LRY
MAA NNN O PCH PSA
REG SPG TCO UDR VNO WRI
0.8
0.8 19
BRE BXP CLI CPT ELS
EQR FRT HCP HIW HME
LRY MAA NNN O PCH
PSA REG SPG VNO
